PrepShorts · Study sheet · Class 8 Mathematics · Chapter 5, Tales by Dots and Lines
Chapter 5 · Tales by Dots and Lines
Which added values move the mean, and in which direction
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One sentence answers this whole topic. A new value drags the average towards itself and never past itself.
The idea
A new value drags the mean towards itself, and never past itself, because all it contributes is its own gap from the old mean — spread over one more share than before. That single sentence answers every question in this stretch of the chapter: which insertions raise the mean and which lower it, why a value equal to the mean changes nothing, how to add two or three values and leave the mean exactly where it was, and why adding the same number to every value lifts the mean by that number rather than by that number times the count. The mean is built from the total, and the total responds to a change in a completely bookkeepable way, so "what will happen to the average?" is a question you can answer before you divide — and sometimes without knowing the individual data at all.
What you should be able to do
- Predict, given the mean of a collection and one new value, whether the mean will rise, fall or stay put, before computing anything
- Compute the new mean after an insertion and check it against that prediction
- Explain why an inserted value equal to the mean leaves the mean unchanged, and say what happens when such a value is removed
- Construct two values whose insertion leaves the mean unchanged, and explain the condition they have to satisfy
- Construct three values that leave the mean unchanged with two of them below the mean, and again with two above it
- State and justify the effect on the mean of adding a fixed number to every value, and of multiplying every value by a fixed number
- Reproduce the chapter's algebraic argument for the additive case, and adapt it to subtraction
- Use the shift rule to correct an average that was computed from measurements carrying a constant error, without re-measuring anything
- Decide when the information given is not enough to say what happens to an average
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| mean | the total of the values divided by how many there are | printed throughout Part II §5.1 (Part II pp.103–108) |
| average | used interchangeably with mean, and the word the algebra pages prefer | printed in Part II §5.1, including both algebraic arguments (Part II pp.107–108) |
| fair-share | the reading of the mean as what each one gets when everything is pooled and split equally | printed in Part II §5.1 (Part II p.105) and again in the SUMMARY (Part II p.133), as the previous class's interpretation |
| dot plot | a number line with one dot per value, stacked where values repeat | printed and used on every page of Part II §5.1 |
| distributive property | the rule that lets a common factor be taken out of a sum | named in the chapter's own multiplication argument (Part II p.108) |
| x₁, x₂, … xₙ | the chapter's notation for a collection of n unnamed values | printed in both algebraic arguments (Part II pp.107–108), read on the printed pages — the subscripts do not survive text extraction |
| gap from the mean | how far a new value sits above or below the current mean, with its direction | an added phrasing; the chapter reasons with this quantity in every case and gives it no name |
| mean-preserving pair | two values whose gaps from the mean cancel, so the mean survives their insertion | an added compound; the chapter sets the task at Part II p.106 and names nothing |
| shift | adding the same fixed number to every value in a collection | an added label; the chapter's heading for this move is "Relatively Unchanged!" (Part II p.106) |
Where people slip up
- "More values means a bigger average." This is printed as a tempting option in Part II p.128 item 6 for exactly that reason. Two more values can raise the average, lower it or leave it alone; what decides is where they sit relative to the current mean, not how many there are.
- "The mean moves to the new value." It moves towards it and stops short. Inserting 17 into a collection averaging 11 moved the mean to 12.2, not to 17, because the newcomer's surplus is shared among all five values.
- "Adding a value below the mean might still raise it." Never. Its gap is negative, so the total gains less than one average share and the mean must fall. This is the "always true" answer to Part II p.115 item 6(ii).
- "Adding 10 to every one of 11 values adds 110 to the mean." It adds 110 to the total, and the total is then shared by 11 again. This is the single most common wrong turn in the section, and the algebra on Part II p.107 exists to close it: 3n/n is 3, not 3n.
- "If the average is wrong I have to measure everything again." The shoes item is built to defeat this. A constant error in every measurement is a constant error in the mean, so it can be removed at the end with one subtraction.
- "Two values that keep the mean must both equal the mean." They only have to have gaps that cancel. There are infinitely many pairs.
- "Doubling the values doubles the mean, so halving them halves the spread and leaves the mean alone." Scaling does both: it multiplies the mean and stretches the distances by the same factor. The relative position of the mean inside the data is what survives, which is what the chapter's heading is pointing at.
- "Every claim about an average can be settled from an average." Item 8 is the counter-case in the other direction: the mean is settled without the group size, and the median is not settled at all. Knowing when the information runs out is part of the skill.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 5.1 Q1, Figure it Out · 5.1 Q3, Figure it Out · 3 Q6, Figure it Out · 3 Q8
Transcript1,387 words
A collection of numbers has an average. Now one more number arrives. What happens to the average? There are three things it could do: go up, go down, or stay exactly where it was. And there is a second question hiding behind the first, which is how far. You could work it out from scratch every time - add everything up again and divide again. But there is one sentence that answers all of it, for any collection and any newcomer, and it is worth having.
Start with four values: five, five, ten and twenty-four. They add up to forty-four, and forty-four over four is eleven. So the average is eleven. Now a fifth value arrives, at seventeen. Add it in. The total is sixty-one, over five values now, and sixty-one divided by five is twelve point two. The average went up, from eleven to twelve point two. It moved by one point two. Where did one point two come from? That is the whole question, and the answer is on the number line rather than in the division.
Look at where the newcomer stood. Seventeen, against an average of eleven. It came in six above. That six is the only thing it brought that the collection did not already have. Everything else about it is ordinary - it takes a share like everyone else. What is new is the surplus. So the direction is not something you observe afterwards. It is settled the moment you see where the newcomer stands.
Above the average, and the average must rise. Below it, and the average must fall. There is no third case and no exception, whatever the other values are doing. Now the size of the move, and this is where guessing goes wrong. The surplus is six. But it does not all land on the average, because the average is shared out. It is shared among everybody - and everybody now includes the newcomer. Five shares, not four.
Six divided by five is one point two. Which is exactly how far the average moved. Try it with four instead and you get one point five, which would make the new average twelve point five. That is wrong, and it is wrong in the most tempting possible way. So here is the sentence. The new average is the old average, plus the newcomer's gap, divided by the new count.
That sentence has a consequence worth saying out loud. The average moves towards the newcomer. It never reaches it, and it never goes past it. Seventeen arrived and the average went to twelve point two - not to seventeen, not even close to seventeen, because that surplus of six had to be split five ways. The bigger the collection, the less any one newcomer can do. Drop seventeen into a hundred values and the average barely twitches.
And taking a value out is the same rule running backwards. Remove something above the average and the average falls: take the twenty-four away and eleven drops to under seven. Which leaves one case. What if the newcomer arrives sitting exactly on the average? Then its gap is zero. There is no surplus to share and no shortfall to cover, so nothing moves. The collection is bigger, the total is bigger, and the average is identical.
It works in both directions too. Take away a value that was sitting on the average and the average does not stir. Now turn the question round. Instead of asking what a newcomer does, ask which newcomers do nothing. Here are seventeen values whose average is nine. Add two more and keep the average at nine. What do they have to be? Not nine and nine - or rather, not only nine and nine.
Five and thirteen works: one is four below, the other four above, and those two gaps cancel before anything is shared. So does a half and seventeen and a half. Among whole numbers from zero to eighteen there are ten such pairs, and they are exactly the pairs that average nine between them. The condition is not that the values are equal to the average. It is that their gaps add up to nothing.
Three at a time, then, and the same rule. Seven, seven and thirteen. Two of them sit two below, one sits four above. Minus two, minus two, plus four - nothing left over. You can see it before you compute it: the two short arrows on one side are exactly paid for by the long one on the other. Now try it the other way round - two newcomers above the average and one below.
Ten and twelve with five works. So does eleven and eleven with five. Among whole numbers up to eighteen there are twenty of them. That is worth noticing. This is not a puzzle with an answer. It is a condition with a family. Different move now. Instead of adding a value, add to every value. Here are eleven numbers. Give every single one of them ten more. The whole plot slides ten to the right, keeping its shape exactly.
So what happens to the average? The most common wrong answer is that the total went up by a hundred and ten, so the average did too. The total did go up by a hundred and ten. Ten for each of the eleven. But then it gets divided by eleven again - the count did not change, because no new values arrived. A hundred and ten over eleven is ten. The average goes up by ten.
In symbols: the new total is the old total plus ten times n, and dividing by n turns ten n into ten. The count cancels itself out. And if you multiply every value instead? Double them all and the total doubles, because the two comes out of every term at once. Then the same count divides it. So the average doubles too. That pair of rules is quietly powerful. Watch.
The first fifty whole numbers average twenty-five and a half. The first fifty odd numbers are each twice a whole number, less one. So their average is twice twenty-five and a half, less one: fifty. The first fifty multiples of four are each four times a whole number. Four times twenty-five and a half is a hundred and two. Two sums nobody did. And a warning while we are here: an average written to two places is usually a rounding. Those eleven values average seven point one eight to two places, but seven point one eight two to three.
Two places these rules earn their keep. Twenty-four students are measured, and the average comes out at a hundred and fifty point two centimetres. Then someone notices they all kept their shoes on, and every shoe adds one centimetre. Nobody needs measuring again. Every value carries the same extra one, so the average carries it too. Subtract one: a hundred and forty-nine point two. Second. Two students join that class of twenty-four, at a hundred and forty-nine and a hundred and fifty-two.
You do not know a single one of the original heights - and you do not need to, because twenty-four times the average gives you their total. The two gaps are minus one point two and plus one point eight, so the average rises. And it rises by about two hundredths of a centimetre. Both halves of that are true. It definitely goes up, and it barely moves. One last thing, and it is about knowing when you have enough.
A group's average weight was sixty-five point three kilograms. This month one person is two kilograms lighter and two people are one kilogram heavier each. Minus two, plus one, plus one. The total has not changed at all, so the average is still sixty-five point three - and notice that nobody ever told you how many people are in the group. You did not need it. Sometimes the information you are missing is information you were never going to use.
And the opposite trap, which is the one to leave with. More values does not mean a bigger average. Two newcomers can push it up, pull it down, or leave it exactly where it stood. What decides is never how many arrive. It is where they stand.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- The mean as the point where the distances balanceClass 8 · Ch 5, Tales by Dots and Lines
- What happens to a product when you nudge one factorClass 8 · Ch 6, We Distribute, Yet Things Multiply
Comes up again in
- Whether adding a value raises or lowers the medianClass 8 · Ch 5, Tales by Dots and Lines